Why do this problem?

This problem gives children a way of sorting numbers according to different properties and also forces them to consider more than one attribute at once. It also provides opportunities for children to explain their placing of the numbers, using appropriate language.

Possible approach

There are two aspects to this problem: Firstly, it focuses on sorting numbers according to certain properties and secondly, it requires a knowledge of how a Carroll diagram works. At first children may find Carroll diagrams challenging to complete because of the need to think about two characteristics simultaneously.

One way to introduce this idea would be simply to ask the children to list the numbers under each of the four headings, written separately and not in the form of a table.

You could then ask questions such as:

What can you tell me about the number $3$?

Why does the number $10$ appear in two lists?

Do all the numbers appear in two lists? Why or why not?

This could then lead on to the Carroll diagram itself. This could either be done with the whole class using the interactivity on an interactive whiteboard, in pairs at individual computers or using these sheets - First and Second. Children should be given time for
discussion both between themselves and as a larger group.

Key questions

What do you know about the number $1$? Where would it go?

What can you tell me about the number ...? Where would it go?

Do you think that the number "$10$" is less than ten?

Where will odd numbers go?

Possible extension

For more of a challenge, you could give children a Carroll diagram with the numbers in place, but with no labels for the cells and ask them to decide what the categories could be. See More Carroll Diagrams for an example.

Possible support

Suggest looking at all the odd numbers first and deciding whether they are less than $10$ or not.

The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the
NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to
embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities
can be found here.